An exact treatment of the space-charge effect in the single-electron analysis of a free-electron laser is presented to calculate its small-signal gain. With the inclusion of the repulsive force between electrons, it is found that the trajectory of an electron can be solved from a generalized equation which includes a space-charge term. The results show the gain is saturated with decreasing growth rate due to high electron density. The radiation frequency is found to increase with the electron density and approach the value at plasma resonance. The condition $\frac{{\omega }_{p}L}{c}=\pi$ clearly defines the boundary between the noninteracting and the collective regime of an electron beam, where ${\omega }_p$ is the plasma frequency, $L$ is the device length, and $c$ is the light velocity in vacuum.

• Received 1 May 1980

DOI:https://doi.org/10.1103/PhysRevA.22.2717